Today is March 14th, or if we write numerically, 3/14th. These are the first three figures of one of the most famous, important and celebrated numbers in science. We are talking about the number π. So today is International Mathematics Day (with a highly recommended website). This year the motto of the celebration is “mathematics for the whole world”, because to understand today’s world we all need to speak the language of mathematics. π is one of the key pieces of the language of mathematics and there are still many open questions about this number.

Much has been written about the number π since the dawn of civilization. And it’s no wonder: the world, as we understand it, depends critically on this universal constant. Its definition is well known: if we take any circle with diameter 1, π is the ratio between its length and its diameter. It is independent of the circle we take: it is a universal geometric invariant. Despite this simple definition, expressing the number π explicitly is very complex. In its decimal expansion there are no obvious numerical patterns. The simplest statistic we can study is the number of occurrences of a given digit.

Experimental results (assuming many digits in the decimal expansion of π) seem to show that each digit appears 10% of the time. But this is just conjecture: as of today it is an open question whether, for any truncation of the decimal expansion of the digits of π, the ratio of each digit is essentially the same.

In addition to the distribution of digits, another interesting question for which the answer is unknown is the following: if we take any number (for example, 44685035261931188171), is it true that this number appears in the decimal expansion of π? All indications and experiments seem to indicate that it is, but so far we don’t have the mathematical tools that allow us to answer this question in a general way.

If the previous result is true, we could say that all universal knowledge exists within π. It goes like this: if we use a numerical encoding method to encode the letters of the alphabet (for example, if we encode the letter a by 00, the letter b by 01, and use the digits 99 to encode the whitespace), any book (written or yet to be written) can be encoded with a very long but finite number. If some pattern is contained in the expansion of pi, we could find Don Quixote, the lost works of Aristotle, and all the novels yet to be written to the end of mankind, even if we had to pick an unimaginable number of numbers.

This seems to indicate that chaos reigns in the expansion of π. However, what is really beautiful is that there are many expressions of the number π where order and patterns reign. Starting with Wallis’ infinite product, a formula proved in 1655 that states that

Or, with sums instead of products, the so-called Leibniz formula, deduced in the 17th century by the German mathematician after whom it is named:

And, let’s not forget what is perhaps the most important formula in mathematics, Euler’s formula, which lists the five most important numbers: 0, 1, the number π, the number e (the base of natural logarithms) and the unit imaginary *you*.

These are just three examples that show the ubiquity of the number π in the mathematical universe and that exemplify the beauty and harmony of mathematics on a day of celebration for our community.

**Juanjo street*** He is an associate professor at the Department of Mathematics at the Polytechnic University of Catalonia (UPC), a member of the Institute of Mathematics at the UPC (IMTech) and a researcher linked to the Mathematics Research Center (CRM).*

**Agate Timon G Longoria*** is the coordinator of the Mathematical Culture Unit at ICMAT*

**Coffee and Theorems**** ***is a section dedicated to mathematics and the environment in which it was created, coordinated by the Institute of Mathematical Sciences (ICMAT), in which researchers and members of the center describe the latest advances in this discipline, share points of contact between mathematics and other social expressions and cultural values and remember those who marked its development and knew how to transform coffee into theorems. The name evokes the definition of the Hungarian mathematician Alfred Rényi: “A mathematician is a machine that turns coffee into theorems.”*

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